A method to calculate pi based on Pythagoras theorem

I have been reading recently a novel about Pythagoras: “Pythagoras killing”, from Marcos Chicot. It depicts a thriller in Pythagoras’ time, where some of the mathematical themes of that era were discussed.

At that time, Pi was not known among the greek. They knew that the relation between the circumference and its diameter was slightly greater than three, but they did not know how to calculate it - even less to understand that it was an irrational number. Moreover, the Pythagorean school thought that all magnitudes in nature came from exact ratios between integer numbers.

In the story, there is a rich man versed in maths that offers a huge amount of gold to whom is able to tell him the ratio between the diameter and the circumference with 4 decimals. The villain (that happens to be a former Pythagoras disciple) wins it with a method based (to add injury to Pithagoras) on the Pythagorean theorem. In principle, these were the only available means to mathematicians of that era to calculate PI - none of the modern methods were available.

A light clicked and I set to calculate PI as well as I could with my Prime. I started doing it with this small graph, and then checked it with a spreadsheet. It is based on calculated more and more sides of the circle, based on the previously calculated side. Good! with 25 iterations, it reaches the 15 decimals available in Excel - the iteration 26 shows 0 difference with Pi.

Will any of you be able to program it with the HP Prime? If you don’t feel comfortable with it, you can try with any other HP calculator, either in RPN or RPL